Homotopy Perturbation Method for Fractional Gas Dynamics Equation Using Sumudu Transform

and Applied Analysis 3 4. Solution by Homotopy Perturbation Sumudu Transform Method (HPSTM) 4.1. Basic Idea of HPSTM. To illustrate the basic idea of this method, we consider a general fractional nonlinear nonhomogeneous partial differential equationwith the initial condition of the form D α t U (x, t) + RU (x, t) + NU (x, t) = g (x, t) , (11) U (x, 0) = f (x) , (12) where Dα t U(x, t) is the Caputo fractional derivative of the function U(x, t), R is the linear differential operator, N represents the general nonlinear differential operator, and g(x, t) is the source term. Applying the Sumudu transform (denoted in this paper by S) on both sides of (11), we get S [D α t U (x, t)] + S [RU (x, t)] + S [NU (x, t)] = S [g (x, t)] . (13) Using the property of the Sumudu transform, we have S [U (x, t)] = f (x) + u α S [g (x, t)] − u α S [RU (x, t) + NU (x, t)] . (14) Operatingwith the Sumudu inverse on both sides of (14) gives U (x, t) = G (x, t) − S −1 [u α S [RU (x, t) + NU (x, t)]] , (15) where G(x, t) represents the term arising from the source term and the prescribed initial conditions. Now we apply the HPM: